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The Poincaré Conjecture has Been Proved
Martin Dunwoody, a famous mathematician who works in the field of topology has a preprint that provides a proof of the Poincaré conjecture. This was one of the seven Clay Mathematics Institute millenium prize problems (reported on Slashdot here). The solution to each of the problems carries a monetary reward of 1 million dollars. However there are a number of conditions that still need to be met for the prize to be awarded in the case of the Poincaré conjecture.
The Poincaré Conjecture proved, and microsoft ads on slashdot
"I think it would be a good idea" Gandhi, on Western Civilisation
so we finally have mathematical proof that a teacup is a donut for every teacup in the known (Euclidean) universe
Without reading the preprint, I cannot say (not that I could understand it anyway :) ). But it wouldn't surprise me if the proof was just for 3.
R^3 is kind of a magical place. R^2 might not have enough wiggling room, but R^4 might have too much. There exists a cross product in only R^3.
Here's the proof:
assume a, b, c such that: a + b = c
then 5a + 5b = 5c
and 4c = 4a + 4b
adding the two: 5a + 5b + 4c = 4a + 4b + 5c
shifting some terms around: 5a + 5b - 5c = 4a + 4b - 4c
simplifying: 5 (a + b - c) = 4 (a + b - c)
dividing by the common factor (a + b - c): 5 = 4
:)
python -c "x='python -c %sx=%s; print x%%(chr(34),repr(x),chr(34))%s'; print x%(chr(34),repr(x),chr(34))"
Still, by the Poincaré Conjecture - Gumby is equivalent to Pokey.
Maybe we should give these problems to the people at the next ACM International Programming Contest.
At the very least, the past participle form should be used, making it "The Poincaré Conjecture has Been Proven"
I've discovered a truly remarkable proof myself. I just can't fit it into this HTML text box.
1) Cows have an even number of legs.
2) Cows have forelegs and two back legs, equalling six legs.
3) Six is an odd amount of legs for a cow.
4) By 1 and 3 cows have both an even number of legs and an odd number of legs.
5) The only number that is both odd and even is infinity.
Cows have an infinite number of legs. QED.
I choose to remain celibate, like my father and his father before him.
Be careful how you phrase that last sentence - your carefree use of the word "obvious" in reference to math calls to mind an old joke:
Two mathematicians were talking one day about some recent work they'd done. One described a proof to the other but quickly glossed over a complicated step. The second one said, "Wait a minute - you didn't prove your last assertion." The reply: "It's obvious."
So the second mathematician wordlessly took a piece of chalk, went to the nearby blackboard, and began to fill it with long statements full of obscure symbols. Nearly half an hour later, he stopped writing, turned around, and said, "You're right. It is obvious."
Here's the better proof that R^3 and S^3 are not homeomorphic. Here it goes:
S^3 is compact and R^3 is not.
Fair enough. But the first one is more likely the correct one, while the second one was added as a concession to the fact that so few could get it right. It's like "data are" and "data is"--both are considered correct now, but literate people use the first.
But the first one is more likely the correct one
Which, I see now, is the one that had been used. Sorry.